The link "Aktuelles" refers to a standard joomla category-blog. The "read more" link is inserted by using JCE Editor. Funny is that different pages with exactly the same setup (headslider, gallery etc.) show different behaviour of the calendar module.

Here are links generated by the module on working/not working pages (month forward button):

judoclub-bauchem.de/news-aktuelles?date=2013-5-1 (ok)
judoclub-bauchem.de/news-aktuelles?id=21&date=2013-5-1 (not ok)

I would like to try uploading your modified file, but I can't download it.

Yes, download works now perfectly.
I uploaded the file and the month forward and back buttons work on all sites
Unfortunately no events show up
Sorry for the inconvenience, I wish I could do more to fix this problem.

OK, I did so. But the Module shows no events neither in v. 2.1.4 standard nor with the modified file you provided. Are there any other caches than the Joomla and the Browser cache (although module caching is set to no)?

I tried to integrate ppgallery instead of sige gallery. for some strange reason it kills the nivo-slider in the head. So I experimented with the JS settings of the plugins. I also uninstalled acymailing, just to eliminate the possibility of a conflict.
Now I can't really say, at which point the events appeared in the module, because I was working on the Gallery/headslider conflict

Yes, I saw that extension but thought, I wouldn't need it. I guess I will give it a try.
Thank you very much for your support. The site I'm working on is for a sports club which means they don't make profit and have not much money. I'm doing this for free because my son is in there, but I appreciate your work and I will definitely suggest them to go pro.

Thank you again